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L=1σ2k 2LBk 2k−2L τL up to some constant, where τLis as defined in Equation (G.22)of Lemma 14 in Section G.2.4. Similar to the proof of Theorem 2, we let Nx=∥x∥0andNy=∥y∥0. Applying Lemma 14 (see Section G below), the right hand side of the above display is bounded by Eh x⊤Hkyi ≲∥x∥∞∥y∥∞2k(Bk)k σ2Sxy B2k+NxNy n2k... | https://arxiv.org/abs/2502.09840v1 |
2.3.16 and Remark 2.3.17 in Tao (2012), we have 1 nEtr(Wk)≲Ck/2(σ2n)k/2 when kis an even integer, and Etr(Wk) = 0when kis odd. When i̸=j, we have E e⊤ iWkej = 0. (E.1) To see this, we take i= 1, j= 2and expand the term e⊤ 1Wke2, Ee⊤ 1Wke2=X j1∈[n]X j2∈[n]···X jk−1∈[n]EWj0j1Wj1j2···Wjk−1jk, where j0= 1andjk= 2. Suppos... | https://arxiv.org/abs/2502.09840v1 |
(F.1). Using the above bound and Equation (F.2), we continue from Equation (F.3) and obtain a⊤ u−u⋆⊤u λ/λ⋆u⋆ ≲max( Blog5n |λ⋆|rµ n∥a∥∞, σp nlog3n |λ⋆|!k0log2n√n) +max Blogn, σ√nlogn |λ⋆|n10, Hao Yan, Keith Levin wheretheinequalityfollowsfromthenaïvebound a+b≤2max{a, b}andthefactthat k0=O(1)inEquation (F.4). Noting t... | https://arxiv.org/abs/2502.09840v1 |
λ⋆ r}defined as D(γ) :=r[ k=1B(λ⋆ k, γ), where B(λ, γ) :={z∈C:|z−λ| ≤γ}. A key step to locate the eigenvalues of Mis to show that in each connected component (in the topological sense) of D(γ), functions fandghave the same number of zeros. This is often established through Rouchés theorem. Theorem 8 (Rouché’s theorem) ... | https://arxiv.org/abs/2502.09840v1 |
example shows that different Jcan correspond to the same graph G, as long as J satisfies the constraints imposed by G. One concrete example of Jsatisfying the above constraints would be j(1)= (1,2,3),j(2)= (4,1,2),j(3)= (2,3,4),j(4)= (3,4,1), so we have EQ4 r=1(ζj(r)−Eζj(r))given by E[(H12H23−EH12H23)(H41H12−EH41H12)(H... | https://arxiv.org/abs/2502.09840v1 |
we have 1≤L≤pk/2by Lemma 6. Lemma 7 shows that for any graph Gconstructed from Rule (i), the terms in Equation (A.3)withJ∈ψ−1(G) all share the same upper bound, and this bound only depends on Gthrough c(G), the number of connected components in G. Lemma 7. For any graph Gconstructed according to Rule (i) with c(G) =L∈[... | https://arxiv.org/abs/2502.09840v1 |
Since we identify G1, G2, . . . , G Lwith the vertices of Gnew, we abuse notation slightly and simply use Gℓto denote the ℓ-th node of Gnew. Whether we intend to refer to a node of Gnewor a connected component of Gwill be clear from the context. By construction, GnewhasLvertices. We connect pairs of vertices in Gnewacc... | https://arxiv.org/abs/2502.09840v1 |
Ireceives a start color, then for some s∈[p],j(r) l=j(s) 1, and thus, its value has Nxpossible choices; if j(r) lreceives an end color, then j(r) l=j(s) k+1for some s∈[p], and its value has Nychoices; if j(r) lreceives an internal color, it has nchoices of values. By counting the number of distinct colors of each type ... | https://arxiv.org/abs/2502.09840v1 |
Γ)is the number of edges in the shortest path from utov. Theundirected distance d0(u, v; Γ)is the number of edges in the shortest path from utovwhen we view the directed graph as an undirected graph. When it is clear from context, we suppress the dependence on Γ and write d(u, v; Γ)andd0(u, v; Γ)asd(u, v)andd0(u, v), r... | https://arxiv.org/abs/2502.09840v1 |
tree Tis either a source or a sink, and thus, every leaf node is a terminal. For t= 1, the claim in Lemma 10 holds trivially since every tree with at least 2nodes has at least 2leaves, which are both terminals. We prove the claim for t≥2by induction. For the base case t= 2, ifTonly has 2terminals, it is a path that can... | https://arxiv.org/abs/2502.09840v1 |
that the edge u0→u1is the only edge connecting T2toT1and there is no outgoing edge from T1toT2. An immediate result is that for any node w∈T1, the path P(w, eT(w);T) does not contain any node in T2. It follows that eT1(w) =eT(w),and d(w, eT1(w);T1) =d(w, eT(w);T). (G.7) Lets2be any source node in T2. Note that for ever... | https://arxiv.org/abs/2502.09840v1 |
than k+ 1nodes, since v1, u1∈T1andu1̸=v1,T1also has at least two leaves. Following the same argument as above, we conclude that there are t′+ 2terminals in T. Finally, if both T1andT2contain at least k+ 1nodes, suppose that |T1|=t1k+l1for1≤l1< k. We have |T2|=β+ 1− |T1| ≥(t′−t1)k+ 2−l1>(t′−t1−1)k+ 2. By the induction h... | https://arxiv.org/abs/2502.09840v1 |
induction hypothesis, Γ\T0accepts at most β− |T0| −(t′−t0+ 1)internal colors. Thus, the forest accepts at most |T0| −t0+β− |T0| −t′+t0−1 =β−t′−1 =β−t distinct internal colors as desired. Otherwise, we have |T0|=β−1,t0=t′+ 1andΓ\T0accepts no internal colors. The forest accepts at most (β−1)−(t′+ 1)< β−tin this case. 2.I... | https://arxiv.org/abs/2502.09840v1 |
¯Gnew l.Take any node v0in¯Gnew l. We will find a source and a sink in ¯Gnew l such that Condition 3 holds for v0. Suppose that (r, ℓ)∈v0for some (r, ℓ)∈ I. We take the start node us containing (r,1)and the end node uecontaining (r, k). By construction of Gnewin Rule (ii), we have d(us, v0;Gnew l) +d(v0, ue;Gnew l)≤k−1... | https://arxiv.org/abs/2502.09840v1 |
not affect the number of internal colors, so the new graph obtained from Operation (I.b) provides a valid upper bound for the number of distinct internal colors. Improved dependence on coherence in eigenvector and eigenvalue estimation error bounds v0u v1 v2w Figure 9: A graphical illustration of the first case after p... | https://arxiv.org/abs/2502.09840v1 |
does not necessarily include all the nodes and edges in the graph. (II.a) For every node w /∈V(P0)with an edge w→v1, we change this edge to w→u. (II.b) Remove the edge v0→v1and mark v1as a start node. An illustration of the resulting graph is shown in Figure 11. Since uandv1share the same first color, Operation (II.a) ... | https://arxiv.org/abs/2502.09840v1 |
it follows from the above display that d seGnew l(u0), u0;eGnew l ≤d v1, u0;eGnew l ≤d s¯Gnew l(u0), u0;¯Gnew l . (G.20) Combining Equations (G.19) and (G.20), we conclude that Condition 3 holds for eGnew l. Step 4. Apply Lemma 11. Finally, by performing Operations (I.a), (I.b), (II.a) and (II.b), we remove one e... | https://arxiv.org/abs/2502.09840v1 |
sequence above are in the same node of Gnew l, so(j(r) l, j(r) l+1)forl∈[k]must all be the same, which further implies that j(r) 1=j(r) 2=···=j(r) k+1. Denote the value of these index as ϕ(0), it is straightforward to see that ϕ(0)∈ {i∈[n] :xi̸= 0, yi̸= 0}, yielding the first bound. Consider 2≤βl< k. Suppose that vsand... | https://arxiv.org/abs/2502.09840v1 |
l∈ Ct⊆[K],tk≥βl≥(t−1)k+ 1 Improved dependence on coherence in eigenvector and eigenvalue estimation error bounds for2≤t≤p/2. By Lemma 13, we obtain the bound for |π−1(Gnew)| p/2Y t=0Y l∈Ct¯τβl= n−2NxNy|C0|nP l∈C0βlp/2Y t=1 n−tNxNt y|Ct|nP l∈Ctβl =nLNxNy n2|C0|p/2Y t=1NxNt y nt|Ct| ,(G.23) where ¯τβlis defined i... | https://arxiv.org/abs/2502.09840v1 |
Testing degree heterogeneity in directed networks Lu Pan∗Qiuping Wang†Ting Yan‡ Abstract We are concerned with the likelihood ratio tests in the p0model for testing degree heterogeneity in directed networks. It is an exponential family distribution on directed graphs with the out-degree sequence and the in-degree seque... | https://arxiv.org/abs/2502.09865v1 |
chi-square approximation when testing the reciprocity parameter in the p1model. Fienberg and Wasserman (1981) suggested a normal approximation for the scaled LRTs under the null that all out-degree parameters (in-degree parameters) are equal to zero. However, these conjectures have not been resolved. Likelihood ratio t... | https://arxiv.org/abs/2502.09865v1 |
there exists a di- rected edge from itoj; otherwise, ai,j= 0. We assume that there are no self-loops, i.e.,ai,i= 0 for all i= 1, . . . , n . Let di=P j̸=iai,jbe the out-degree of node iand d= (d1, . . . , d n)⊤the out-degree sequence of the graph Gn. Similarly, let bj=P j̸=iai,j be the in-degree of node jandb= (b1, . .... | https://arxiv.org/abs/2502.09865v1 |
j̸=ieα(k+1) i+β(k+1) j 1+eα(k+1) i+β(k+1) j/di, i=r+ 1, . . . , n , zj=P j̸=ieα(k+1) i+β(k+1) j 1+eα(k+1) i+β(k+1) j/bj, j= 1, . . . , n 10:Ifmax(max i(|yi−1|),max j(|zj−1|))> ϵthen 11: Turn to line 3 12:end for 13:Ifmax(max i(|yi−1|),max j(|zj−1|))< ϵthen 14: Break 15:end for 16: k=k+ 1 17:end while 18: Return ˆ α=α(k... | https://arxiv.org/abs/2502.09865v1 |
difference of two distributions of¯bn v2n,2nand¯bn−Pr i=1¯di ˜v2n,2nis much larger than the order of log n/n. Under the homogenous null, the off-diagonal elements are the same as the approximate inverse for approximating V−1in the full parameter space. This means that the difference between the two terms above is not c... | https://arxiv.org/abs/2502.09865v1 |
quantiles agree well with theoretical quantiles except for there are only slight deviations from the reference line y=xin the tail of curves for the normalized LRT. We also record the Type I errors under two nominal levels τ1= 0.05 and τ2= 0.1, shown in Table 1. From this table, we can see that the simulated Type I err... | https://arxiv.org/abs/2502.09865v1 |
to increase with cand approaches 100% when c= 1.3. When nandcare fixed while rincreases, a similar phenomena can be seen. The power tends to grow with nincreases when candrare fixed. 3.2 Testing degree heterogeneity in real-world data We apply likelihood ratio statistics to test degree heterogeneity in three real-world... | https://arxiv.org/abs/2502.09865v1 |
degree heterogeneity in a large set of nodes. When rincreases to 400, the p-value becomes significant. Further, the p-values under r= 500 is much less than those under r= 400. These results show the degree heterogeneity is a common phenomenon and varies across different sets of nodes 11 in different data sets. Table 3:... | https://arxiv.org/abs/2502.09865v1 |
2n−1,(9) whereev2n,2n=P2n−1 i=r+1evi,2n=Pn i=r+1vi,2n+P2n−1 i=n+1Pr j=1vi,jandδi,jis the Kronecker delta function. Yan et al. (2016) proposed to use the matrix Sto approximate V−1. Lemma 1. ForV∈ L n(1/bn,1/cn)withn≥2and its bottom right (2n−1−r)×(2n−1−r) block V22withr∈ {1, . . . , n −1}, if (n−r)m (n−1)rM=o(1), we ha... | https://arxiv.org/abs/2502.09865v1 |
−n(n−r)¯b3 n cnv3 2n,2n,(15) nX i=r+1n−1X j=1,j̸=i(bαi−αi)2(bβj−βj)µ′′(πij) =Op(1−r/n)b11 n(logn)2 c8 n −n(n−r)¯b3 n cnv3 2n,2n,(16) nX i=r+1n−1X j=1,j̸=i(bαi−αi)(bβj−βj)2µ′′(πij) =Op(1−r/n)b11 n(logn)2 c8 n +n(n−r)¯b3 n cnv3 2n,2n.(17) Ifbαiis replaced with bα0 ifori=r+ 1, . . . , n , then the above upper bound st... | https://arxiv.org/abs/2502.09865v1 |
explicit expressions of bθandbθ0. Step 2 is about the explicit expression of B1−B0 1. Step 3 is about showing that the main term involved with B1−B0 1asymptotically follows a normal distribution and the remainder terms goes to zero. Step 1. We characterize the asymptotic representations of bθandbθ0. Recall that πij=αi+... | https://arxiv.org/abs/2502.09865v1 |
b n−1)⊤andeVdenote the Fisher information matrix ofeθ= (α1, αr+1, . . . , α n, β1, . . . , β n−1)⊤under the null H0:α1=···=αrwith r≤n, where eV= ˜v11˜v⊤ 12 ˜v12V22! , (35) where V22is the lower right (2 n−1−r)×(2n−1−r) block of V,˜v12= (˜v1,r+1, . . . , ˜v1,2n−1)⊤, and ˜v11=X j̸=1eα1+βj (1 +eα1+βj)2+···+X j̸=reα1+βj (1... | https://arxiv.org/abs/2502.09865v1 |
and eh= (˜h1,˜hr+1, . . . , ˜h2n−1)⊤satisfies |˜h1|≲rb6 nlogn c5n, max i=r+1,...,2n−1|˜hi|≲b6 nlogn c5n.(41) With some ambiguity of notations, we still use the notation ehhere that is a little different from ehdefined in (31) in Section 5.1. Specifically, the first element of ehcan be viewed as the sum of ehi,i= 1, . .... | https://arxiv.org/abs/2502.09865v1 |
central limit theorem for bounded case [Lo´ eve (1977) (p.289)], ˜ v−1/2 11Pr i=1¯diconverges in distribution to the standard normal distribution if ˜v11→ ∞ . Therefore, as r→ ∞ , [Pr i=1{di−E(di)}]2/˜v11 r=op(1). By combining (44), (45), (46) and (47), it yields 2(B1−B0 1)√ 2r=1√ 2rrX i=1(di−Edi)2 vi,i+op(1). Therefor... | https://arxiv.org/abs/2502.09865v1 |
proof of Theorem 1 (a), it is sufficient to demonstrate: (1) 2( B1−B0 1) converges in distribution to the chi-square distribution with r−1 degrees of freedom; (2) B2−B0 2=Opb21 n(logn)5/2 n1/2c17 n , B 3−B0 3=Opb18 n(logn)5/2 n1/2c14 n . These two claims are proved in the following three steps, respectively. Step 1... | https://arxiv.org/abs/2502.09865v1 |
i=n+1hi)12n−r−1h2| ≲b18 n(logn)2 nc15 n+ (2n−r)·bn n·b3 n n2c3 n·b6 nlogn c5 n2 (56) ≲b18 n(logn)2 nc15 n. In view of (37) and (53), the upper bounds of C2andC3are derived as follows: |C2|=|(h2−eh2)⊤fW22h2| ≤(2n−r)2∥fW22∥max∥h2−eh2∥∞∥h2∥∞ ≲n2·b3 n n2c2 n·b27 n(logn)2 n1/2c9 n·b6 nlogn≲b36 n(logn)3 n1/2c11 n. (57) 28 ... | https://arxiv.org/abs/2502.09865v1 |
Self-Normalized Inference in (Quantile, Expected Shortfall) Regressions for Time Series∗ Yannick Hoga†Christian Schulz† February 17, 2025 Abstract This paper is the first to propose valid inference tools—based on self-normalization—in time series expected shortfall regressions. In doing so, we propose a novel two-step ... | https://arxiv.org/abs/2502.10065v1 |
such time series ES regressions suffer from several drawbacks that severely restrict their usefulness. For instance, the i.i.d. frame- work of Dimitriadis and Bayer (2019) is unsuitable as macroeconomic data typically exhibit strong serial dependence. Similarly, the predictive ES models of Patton et al. (2019) impose a... | https://arxiv.org/abs/2502.10065v1 |
covariance and the bootstrap approach. The former requires the choice of several tuning parameters: the estimation of the bread matrix in the long-run variance depends on a bandwidth choice for estimating the scalar sparsity (Kato, 2012) and the meat matrix requires a bandwidth to be used in the kernel-weighted estimat... | https://arxiv.org/abs/2502.10065v1 |
6 concludes. The proofs of the main theoretical results are relegated to the Appendix. 2 Quantile Regressions Since our two-step approach to estimating ES regressions (to be introduced in Section 3) relies on an initial QR estimate, we first consider time series QRs in this section. The theory will then be used later o... | https://arxiv.org/abs/2502.10065v1 |
bandwidth requires the estimation of unknown quantities. The second available approach to inference in time series QR—the bootstrap approach of Gre- gory et al. (2018)—likewise involves the choice of tuning parameters, viz. the bandwidth for the smoothing kernel estimator and the block length. To sidestep such problems... | https://arxiv.org/abs/2502.10065v1 |
with the Skorohod topology (Billingsley, 1999). Theorem 1. Under Assumptions 1–3 it holds for any ϵ∈(0,1)that, as n→ ∞ , s√nbα(s)−α0d−→Ω1/2Wk(s) inDk[ϵ,1], where Wk(·)is ak-variate standard Brownian motion and Ω=D−1JD−1. Proof: See Appendix A. For the non-functional case with s= 1, Theorem 1 is similar to Theorem 2.2... | https://arxiv.org/abs/2502.10065v1 |
Davidson, 1994, Theorem 26.13) to Theorem 1 essentially implies the following corollary. Corollary 1. LetAbe a full rank (ℓ×k)-matrix with ℓ≤k, and define α0,A=Aα 0and bαA(s) =Abα(s). Under the assumptions of Theorem 1 it holds that, as n→ ∞ , Tn:=nbαA(1)−α0,A′S−1 n,bαA(·)bαA(1)−α0,Ad−→W′ k(1)A′V−1 kAW k(1) = :W(A)... | https://arxiv.org/abs/2502.10065v1 |
as it allows the ES regression in (3) to be estimated. Often, the parameters in joint (quantile, ES) models are estimated by exploiting elicitability of the pair (quantile, ES) with appertaining loss functions given in Fissler and Ziegel (2016); see, e.g., Dimitriadis and Bayer (2019) and Patton et al. (2019). However,... | https://arxiv.org/abs/2502.10065v1 |
pre-multiplication by (the inessen- tial)−1/2, the ES generalized errors are ψ∗(εt, ξt) =−(1/2)∂µLES (v, µ), y (v,µ,y )=(Qτ(Yt|Xt),ESτ(Yt|Xt),Yt) =1{y>v}(y−µ) (v,µ,y )=(Qτ(Yt|Xt),ESτ(Yt|Xt),Yt) =1{εt>0}ξt. Similarly as the QR generalized errors satisfy Et ψ(εt) = 0, we also have that Et ψ∗(εt, ξt) = 0. Moreover, ... | https://arxiv.org/abs/2502.10065v1 |
see the matrix A0in their Theorem 2. Therefore, their modeling approach is well-suited for predictive modeling of quantiles and ES, but may lead to invalid inference in time series ES regressions. 12 4 Simulations 4.1 The Data-Generating Process We consider a time series regression as a data-generating process (DGP), w... | https://arxiv.org/abs/2502.10065v1 |
its moving block design, and mitigates issues related to the smooth conditional density of the errors inDby employing a two-fold smoothing strategy: tapering for data blocks and kernel smoothing for individual observations. Therefore, the HAC and the SBB approaches rely on a data-driven estimation of the bandwidth, and... | https://arxiv.org/abs/2502.10065v1 |
4.8 4.4 13.0 8.3 5.0 4.8 0.9 38.7 9.2 26.2 4.5 36.8 9.3 23.4 5.0 34.6 11.2 20.2 5.5 Notes: We compute the i.i.d. errors using the iidoption from the rqfunction in the quantreg package in R. For the HAC standard errors, the data-driven optimal bandwidth, bS∗ T, is calculated by the alternative autocorrelation estimator ... | https://arxiv.org/abs/2502.10065v1 |
H0:β0,2=δ◦ 2+η2ESτ(et), (8) with δ◦ 2= 1 and η2= 0.5. For the quantile regressions, we were able to compare our SN-based test against several alternative methods. However, for the ES regressions, no comparable approaches allowing for autocorrelated errors are available. Therefore, Table 2 only shows size for our SN-bas... | https://arxiv.org/abs/2502.10065v1 |
also provide an empirical illustration for our results on quantile regressions (Section 5.1). We do so, because for QR our method can be compared to the bootstrap-based approach of Gregory et al. (2018) and the direct estimation of the asymptotic variance-covariance matrix of Galvao and Yoon (2024). Such a direct compa... | https://arxiv.org/abs/2502.10065v1 |
Return 0.0 0.5 6.80.9 3.4 16.1 0.2 0.2 0.0 Stock Variance 24.4 55.4 4.7 2.2 0.6 12.0 4.7 45.9 1.5 Notes: We use 10 lags in the specification of the DQT. p-values ≤5% in bold. available data from 1927 to 2023 for all four regressors.1This gives us a sample of size n= 1,163. We estimate the parameters α0= (α0,1, α0,2)′of... | https://arxiv.org/abs/2502.10065v1 |
as our SN-based approach. Figure 4 presents the estimates of α0,2from (9) for each predictor over different quantile levels ranging from 0.1 to 0.9. The colored lines around the estimates are confidence intervals based on the IID, HAC, SBB, and self-normalized approach. Since we only test one coefficient at a time, Sn,... | https://arxiv.org/abs/2502.10065v1 |
inference. Further, and more importantly, in order to make statements about the conditional ES of GDP growth, we do not have to rely on the fitted skewed- tdistribution but can directly estimate the effect of the NFCI on the ES of the GDP growth distribution. We then test the statistical significance of the predictors ... | https://arxiv.org/abs/2502.10065v1 |
(2019, Figure 4, Panel C) and—as a novelty—adds SN-based confidence intervals to it. Note that Adrian et al. (2019) do not provide such or other autocorrelation-robust confidence intervals, e.g., those of Gregory et al. (2018) or Galvao and Yoon (2024). Figure 5 shows that the regression slopes change for the NFCI acro... | https://arxiv.org/abs/2502.10065v1 |
the equity premium using data from 1927 to 2023. The SN-based approach proves to be most robust for the majority of the considered cases. Second, we revisit Adrian et al. (2019)’s seminal work in the Growth-at-Risk literature. Adrian et al. (2019) do not statistically test the effect of lagged financial conditions on t... | https://arxiv.org/abs/2502.10065v1 |
implies (on a Skorohod probability space) the uniform convergence sups∈[a,b] fn(xi, s)−f∞(xi, s) −→0 almost surely for each i[the equivalent of Kato’s equation (4)]. This conclusion holds because Billingsley (1999) shows that when convergence inDℓ[a, b] is to a continuous limit [here: the f∞(xi,·)’s in (A.1)] convergen... | https://arxiv.org/abs/2502.10065v1 |
/ncan easily be shown to be negligible. It follows that Sn,bαA(·)=1 n2nX j=⌊nϵ⌋+1j2bαA(j/n)−bαA(1)bαA(j/n)−bαA(1)′ ≤nZ1 ϵs2bαA(s)−bαA(1)bαA(s)−bαA(1)′ds ≤1 n2nX j=⌊nϵ⌋+1(j+ 1)2bαA(j/n)−bαA(1)bαA(j/n)−bαA(1)′, where the inequalities are to be understood with respect to the Loewner order ≤for matrices. Thus, ... | https://arxiv.org/abs/2502.10065v1 |
=J1/2 ∗W∗(s)+KΩ1/2W(s) andB(s) =s(1−τ)D∗. Similarly as in the proof of Theorem 1, we deduce from this that Wn(w1, s), . . . , W n(wℓ, s)′d−→ W∞(w1, s), . . . , W ∞(wℓ, s)′inDℓ[ϵ,1] for any w1, . . . ,wℓ∈Rk, such that (A.1) of Lemma 1 is met. Once more, the only remaining condition of (i)–(iii) in Lemma 1 that is n... | https://arxiv.org/abs/2502.10065v1 |
uses that J22J′ 22=J−C21C′ 21=J−C(J−1/2 ∗)′J−1/2 ∗C′=J−CJ−1 ∗C′from (B.4). We obtain that Cov fW(s),fW(t) = min {s, t}Σ∗, which characterizes Brownian motion (e.g., Jacod and Shiryaev, 1987, Theorem 4.4 II), such that fW(s)d=Σ1/2 ∗Wk(s) for a k-dimensional standard Brownian motion Wk(·). We conclude from (B.1) that w∞... | https://arxiv.org/abs/2502.10065v1 |
1 nnX t=1∥Xt∥2=OP(1). (B.14) Combining (B.8) with (B.13) and (B.14), we obtain that sups∈[0,1] V2n(w, s) =oP(1). To prove (B.11), write V1n(w, s) =⌊ns⌋X t=1 1{w′n−1/2Xt<0}fεt|Xt(0)1 2(w′n−1/2Xt)2 +1{w′n−1/2Xt>0}fεt|Xt(0)1 2(w′n−1/2Xt)2 =1 2w′1 n⌊ns⌋X t=1fεt|Xt(0)XtX′ t w. Since fεt|Xt(0) is a Xt-measurable random v... | https://arxiv.org/abs/2502.10065v1 |
|ν2t(w)−ν2t(w)|r ≤Kn n−r+1 2E ∥Xt∥r+1 +n−rE ∥Xt∥2ro ≤Kn−r+1 2, (B.20) 39 where we used Assumption 3 (i) for the final inequality. In particular, ∥ν2t(w)−ν2t(w)∥r≤Kn−1+1/r 2, (B.21) E |ν2t(w)−ν2t(w)|2 ≤Kn−3/2, (B.22) where the latter inequality follows because (B.20) holds not just for r >2 from Assumption 2, but... | https://arxiv.org/abs/2502.10065v1 |
(1994, Corollary 29.19). To do so, we verify the assumptions of that corollary. First, note that the {1{εt>0}ξtXt}have zero mean, because Et 1{εt>0}ξt =Pt{εt>0}1 Pt{εt>0}Et 1{εt>0}ξt =Pt{εt>0}Et ξt|εt>0 (5)= (1 −τ)Et ξt|ξt> Q τ(ξt|Xt) = (1 −τ) ES τ(ξt|Xt) = 0 , such that by the LIE, E 1{εt>0}ξtXt =Eh Et 1{εt... | https://arxiv.org/abs/2502.10065v1 |
by showing that sup s∈[ϵ,1] 1 n⌊ns⌋X t=11{X′ t(bα(s)−α0)<εt≤0}XtX′ t P−→0 and sup s∈[ϵ,1] 1 n⌊ns⌋X t=11{0<εt≤X′ t(bα(s)−α0)}XtX′ t P−→0. For brevity, we only prove the latter convergence. Write P sup s∈[ϵ,1] 1 n⌊ns⌋X t=11{0<εt≤X′ t(bα(s)−α0)}XtX′ t > ε =P sup s∈[ϵ,1] 1 n⌊ns⌋X t=11{0<εt≤n−1/2X′ t√n(bα(s)−α0)}XtX′ t >... | https://arxiv.org/abs/2502.10065v1 |
sup s∈[0,1] 1 n⌊ns⌋X t=1n XtX′ t(α0−β0)X′ tfεt|Xt(0)−E XtX′ t(α0−β0)X′ tfεt|Xt(0)o =oP(1). With this and the observation that E XtX′ t(α0−β0)X′ tfεt|Xt(0)(5)=E (ξt−εt)fεt|Xt(0)XtX′ t , we obtain using Assumption 3* (iv) that sup s∈[0,1]sup ∥v∥≤K V1n(v, s)−sKv ≤sup s∈[0,1]sup ∥v∥≤K V1n(v, s)−1 n⌊ns⌋X t=1E XtX′ t(... | https://arxiv.org/abs/2502.10065v1 |
the second step follows from the LIE, the third step from (5), the fourth step from the cr- inequality, the third-to-last step from Assumption 1 (ii) and the last step from Assumption 3* (i). For the other expectation on the right-hand side of (B.36), similar arguments show that E |ν2t(v)|r =n−r/2Eh Et{1{0<εt≤v′n−1/2... | https://arxiv.org/abs/2502.10065v1 |
t=1|Xt|Et (1{0<εt≤(ℓ+2)ρn−1/2Xt}−1{0<εt≤ℓρn−1/2Xt})|ξt| , (B.39) where the maximum is taken over the integers ℓ, such that [ ℓρ,(l+ 2)ρ]⊂[−K, K ]. Now, Et (1{0<εt≤(ℓ+2)ρn−1/2Xt}−1{0<εt≤ℓρn−1/2Xt})|ξt| ≤Et 1{ℓρn−1/2Xt<εt≤(ℓ+2)ρn−1/2Xt}|ξt| =Et 1{ℓρn−1/2Xt<εt≤(ℓ+2)ρn−1/2Xt}|Xt(α0−β0) +εt| ≤ |α0−β0| · |Xt|Et 1{ℓρ... | https://arxiv.org/abs/2502.10065v1 |
M. (2024). Capturing macro-economic tail risks with Bayesian vector autoregressions. Journal of Money, Credit and Banking , 56(5):1099–1127. Cenesizoglu, T. and Timmermann, A. (2008). Is the distribution of stock returns predictable? SSRN Preprint , pages 1–50. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=110718... | https://arxiv.org/abs/2502.10065v1 |
continuous treatment effects. Forthcoming in the Journal of Business & Economics Statistics (https://doi.org/10.1080/07350015. 2024.2430293 ), pages 1–31. Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Le Cam, L. M. and Neyman, J., editors, Proceedings of the Fifth Be... | https://arxiv.org/abs/2502.10065v1 |
parameter stability in quantile regression models. Statistics & Probability Letters , 78(16):2768–2775. Wang, R. and Shao, X. (2020). Hypothesis testing for high-dimensional time series via self-normalization. Annals of Statistics , 48(5):2728–2758. Wei, B., Tan, K. M., and He, X. (2024). Estimation of complier expecte... | https://arxiv.org/abs/2502.10065v1 |
arXiv:2502.10143v2 [math.ST] 9 Apr 2025Zeropatternsinmulti-waybinarycontingencytableswithuniform margins RobertoFontanaa,ElisaPerroneb,FabioRapalloc aDepartmentofMathematicalSciences,PolitecnicodiTorin o,CorsoDucadegliAbruzzi24,10129Torino,Italy bDepartmentofMathematicsandComputerScience,Eindhoven UniversityofTechnolog... | https://arxiv.org/abs/2502.10143v2 |
(474,4229), while the defininggrids ofthe corresponding discrete copu la are/u1D4482=1 /u1D441{0,˜/u1D4621,˜/u1D4621+˜/u1D4622}= {0,0.12,1}and/u1D4492={0,0.1,1}. Theentriesofthediscretecopula /u1D4361=/u1D436/u1D4482,/u1D4492=(/u1D450/u1D456,/u1D457),/u1D456∈ [2]and/u1D457∈ [2] arecomputedfromtheentriesofthecontingencyt... | https://arxiv.org/abs/2502.10143v2 |
Weobservethatifthezeropatternisoneofthoseexpressedb ytheconditionofProposition3.1,e.g., bychoosing /u1D4561=1,/u1D4562=2,/u1D466/u1D4561=0,/u1D466/u1D4562=0thatmeans /u1D45D/u1D6FC=/braceleftBigg 0/u1D6FC=(0,0,/u1D6FC′′)or/u1D6FC=(1,1,/u1D6FC′′), /u1D6FC′′∈/u1D449/u1D451−2 >0 elsewhere. wecanbuildacompatibletable /u1D4... | https://arxiv.org/abs/2502.10143v2 |
01 40 0 0 0 1 1 0 0 0 01 21 400 0 1 1 0 0 1 0 0 01 201 40 0 1 1 1 0 1 0 01 201 400 0 1 0 0 1 1 01 20 01 400 0 1 1 1 1 11 20 0 0 01 40 0 1 0 Table3:/u1D451=3:Extremepmfswithuniformmarginsandfourzeropatterns. 11 Firsttable (/u1D44B4=0) ( /u1D44B4=1) Secondtable (/u1D44B4=0) ( /u1D44B4=1) Figure2: Two24tableswithzeropatte... | https://arxiv.org/abs/2502.10143v2 |
(0,0)inthestrictlypositivesimplexoftheprobabilities.Thel ocalbehaviorof /u1D6FD(/u1D6FC)canbe studiedbymeansoftheimplicitfunctionstheorem.Thepart ialderivativesare /u1D715/u1D453 /u1D715/u1D6FC=˜/u1D45D111−˜/u1D45D001−2/u1D6FC,/u1D715/u1D453 /u1D715/u1D6FD=˜/u1D45D111+˜/u1D45D001+/u1D71412 1(˜/u1D45D011+˜/u1D45D101) +2... | https://arxiv.org/abs/2502.10143v2 |
Proportional Fit ting, John Wiley & Sons, Ltd, 2006. doi:https://doi.org/10.1002/0471667196.ess1297.pub2 . [18] S. E. Fienberg, A. Rinaldo, Maximumlikelihoodestimat ioninlog-linearmodels, Ann. Stat. 40 (2) (2012) 996–1023. doi:10.1214/12-AOS986 . URLhttps://doi.org/10.1214/12-AOS986 [19] J.I.Coons,C.Langer,M.Ruddy,Clas... | https://arxiv.org/abs/2502.10143v2 |
arXiv:2502.10161v1 [stat.ME] 14 Feb 2025Revisiting the Berkeley Admissions data: Statistical Test s for Causal Hypotheses Sourbh Bhadane1, Joris M. Mooij1, Philip Boeken1, and Onno Zoeter2 1Korteweg-de Vries Institute for Mathematics, University o f Amsterdam 2Booking.com, Amsterdam, The Netherlands February 17, 2025 A... | https://arxiv.org/abs/2502.10161v1 |
while the causal graph is akin to a simple mediation graph with sex being the treatment, department choice being the mediat or and the admissions decision be- ing the outcome. However, in both Pearl (2009) and Pearl and M ackenzie (2018), Pearl notes that merely conditioning on department choice might not always be app... | https://arxiv.org/abs/2502.10161v1 |
other analogous settings, e.g. to investigate sex d iscrimination in awarding distinctions to PhD students (Bol, 2023). 1Some directly inspired by the Berkeley admissions case, for example the path-dependent counterfactual fairness notion in Kusner et al. (2017, Appendix S4). 2Albeit possibly post-selection, which we d... | https://arxiv.org/abs/2502.10161v1 |
kind w e allow for in the Berkeley dataset. 2 Preliminaries We outline a few definitions that follow the formal setup of Bonge rs et al. (2021). Definition 1 (Structural Causal Model (SCM)) .AStructural Causal Model (SCM) is a tuple M= (V,W,X,f,P)where a) V,W are disjoint, finite index sets of endogenous and exogenous ran... | https://arxiv.org/abs/2502.10161v1 |
the view that modeling assump- tions describe a family of SCMs and fairness notions define a sub set of this family. We relate ex- isting general notions of fairness in the literature to this v iewpoint. While this is a re-examination of the various existing analyses of the Berkeley admissions c ase, in the next section... | https://arxiv.org/abs/2502.10161v1 |
proposed this notion for the Berkeley dat a. A valid test for this notion is a conditional independence test for A⊥ ⊥S|D. Indeed, the analysis of Bickel et al. (1975) shows that the data contain not enough evidence to reject the null h ypothesis that this conditional independence holds, and therefore, concludes fairnes... | https://arxiv.org/abs/2502.10161v1 |
same. Definition 11 (Counterfactual Notion of Fairness) .M∈Mno-cf is fair according to the counter- factual notion of fairness if it belongs to the null hypothesis set H0 no-cf-ctrf/defines/braceleftBig M∈Mno-cf:∀d,s,PM(Ado(S=s,D=d)=Ado(D=d)) = 1/bracerightBig . The alternate hypotheses are given by the complement of th... | https://arxiv.org/abs/2502.10161v1 |
then we can conclude that the underlying causal model does n ot belong to the causal null hypothesis of any of the fairness notions, i.e., there is unfairness. In the next section, we enlarge the class of models to allow for confounding between Dand Aand perform a similar reasoning exercise. 4 Berkeley Case: With Laten... | https://arxiv.org/abs/2502.10161v1 |
absence of edges, impose equality or inequality constraints (Evans, 2016; Wolfe et al ., 2019) in addition to conditional in- dependence constraints which are the only constraints impo sed by a DAG. For the Berkeley 5Since this allows for causal cycles, this would require usin g the framework of simple SCMs (Bongers et... | https://arxiv.org/abs/2502.10161v1 |
fair, i.e., our conclusion should be that fairness is undecid- able. In Section 5 we introduce a Bayesian test for the IV inequ alities. 4.2 Bounds on Interventional Notion of Fairness ForM∈Mcf+, the interventional notion of fairness is the CDE, which is no t identifiable in our case. By a response-function parameteriza... | https://arxiv.org/abs/2502.10161v1 |
fairness notions for a particular case with respect to the same causal modeling assumptions. In this section, we consider the three existing counterfactual fairness notions , namely the NDE (Nabi and Shpitser, 2018; Chiappa, 2019), and the counterfactual and path-depen dent counterfactual fairness no- tions in Kusner e... | https://arxiv.org/abs/2502.10161v1 |
Dirichlet distribution. Using n= 106samples, we ob- serve no violations of the IV inequality. Therefore, the confi dence interval for the posterior prob- ability of the Berkeley data satisfying the IV inequalities is/bracketleftbig 1−3.69×10−6,1/bracketrightbig . As mentioned in Section 4.1 satisfying the IV inequalitie... | https://arxiv.org/abs/2502.10161v1 |
and Arvind Narayanan. Fairness and machine learning: Limita- tions and opportunities . MIT press, 2023. Richard A Berk, Arun Kumar Kuchibhotla, and Eric Tchetgen Tc hetgen. Fair risk algorithms. Annual Review of Statistics and Its Application , 10(1):165–187, 2023. Peter J Bickel, Eugene A Hammel, and William J O’Conne... | https://arxiv.org/abs/2502.10161v1 |
in the fair de- termination of risk scores. In 8th Innovations in Theoretical Computer Science Conference , 2017. Matt J Kusner, Joshua Loftus, Chris Russell, and Ricardo Silv a. Counterfactual fairness. Advances in Neural Information Processing Systems , 30, 2017. Razieh Nabi and Ilya Shpitser. Fair inference on outco... | https://arxiv.org/abs/2502.10161v1 |
fact, IV inequalities are more appropriately expressed in terms of PM(X,Y|do(Z)). Lemma 22. LetMIV/defines{M:G(M)is a subgraph of Figure 1}. For any M∈MIV, max x/summationdisplay ymax zPM(X=x,Y=y|do(Z=z))≤1. (4) Proof. Since PM(X=x,Y=y|do(Z=z)) =PM(fX(z,U) =x,fY(x,U) =y) ≤P(fY(x,U) =y) =PM(Y=y|do(X=x)), max x/summation... | https://arxiv.org/abs/2502.10161v1 |
identical to (4). Further, PMIV+={PM(Z) :M∈MIV+}⊗KMIVsince assuming positivity, MIV=MIV+and PM(X,Y|do(Z)) =PM(X,Y|Z)forM∈MIV+. Since the first factors are identical, Theorem 17 follows from Lemma 23. Proof of Lemma 23. ForM∈MIV, the response-function parameterization yields a counter- factually equivalent SCM (Forré and... | https://arxiv.org/abs/2502.10161v1 |
inequality” , and say that two such entries overlap if i=j. We also call the set of entries of b corresponding to K(X,Y|Z=z)a “stratum” . We proceed case by case. 20 1. More than two IV inequalities are active in b. This yields a contradiction with the normal- ization constraints. 2. Exactly two IV inequalities are act... | https://arxiv.org/abs/2502.10161v1 |
=fA(S,d,UA,U)) = 1, (9) 21 implying that M∈H0 cf-ctrf. For the converse, M∈H0 cf-ctrfimplies (9). For s/\e}atio\slash=s′, and all d, PM(fA(s,d,UA,U) =fA(S,d,UA,U)) =PM/parenleftbig fA(s,d,UA,U) =fA(s′,d,UA,U)/parenrightbig PM/parenleftbig S=s′/parenrightbig +PM(S=s). IfPM(S=s′)>0, we conclude PM(fA(s,d,UA,U) =fA(s′,d,U... | https://arxiv.org/abs/2502.10161v1 |
we will also refer to them a s collection of response- function-parameterized SCMs. From interventional equivalence (which follows as a result o f counterfactual equivalence) of the response-function-parameterization, we have Kcf-graph=/braceleftBig P˜M(D,A|do(S)) :˜M∈¯H0 graph/bracerightBig Kcf-inter=/braceleftBig P˜M... | https://arxiv.org/abs/2502.10161v1 |
conditional i ndependence test A⊥ ⊥S|D. Definition 29 (Path-dependent Counterfactual Fairness (Kusner et al., 20 17)) .M∈Mno-cf is fair if for alls,d,PM(s,d)>0implies PM/parenleftBig Ado(S=s′,D=d)= 1|D=d,S=s/parenrightBig =PM/parenleftBig Ado(S=s,D=d)= 1|D=d,S=s/parenrightBig fors′/\e}atio\slash=s. Proposition 30. /brac... | https://arxiv.org/abs/2502.10161v1 |
path-dependent counterfactual fairness then PM(D,A,S)∈ Pcf-graph=PIV+. IfM∈H0 cf-graph +, thenMsatisfies path-dependent counterfactual fairness. Proof. We show that a model M∈Mcf+that satisfies the path-dependent counterfactual fair- ness notion is observationally equivalent to a model in H0 cf-graph +. This implies that... | https://arxiv.org/abs/2502.10161v1 |
posterior, P(θ| R1,R2,···Rm)which is also a Dirichlet distribution. Using n= 106samples, we observe no vio- lations of the IV inequality. Therefore, the confidence inter val for the posterior probability of the cum-laude data satisfying the IV inequalities is/bracketleftbig 1−3.69×10−6,1/bracketrightbig . In contrast to... | https://arxiv.org/abs/2502.10161v1 |
Bayesian calculus and predictive characterizations of extended feature allocation models Mario Beraha1, Federico Camerlenghi1, and Lorenzo Ghilotti1 1Department of Economics, Management, and Statistics, University of Milano–Bicocca, 20126 Milano, Italy Abstract We introduce and study a unified Bayesian framework for ex... | https://arxiv.org/abs/2502.10257v2 |
(Stolf and Dunson, 2024), and microbiome studies (James et al., 2025). See also Griffiths and Ghahramani (2011) for further discussions. Masoero et al. (2022) recently proposed to employ feature allocation models to study the so- calledunseen feature problem : given a sample of nobservations displaying Knunique feature... | https://arxiv.org/abs/2502.10257v2 |
2003; Lijoi et al., 2007) as the only normalized 3 completely random measure (Regazzini et al., 2003) that is also a Gibbs-type prior. Additional characterizationsexistforneutral-to-rightpriors(MuliereandWalker,1999)andMarkovchains (Zabell, 1995; Rolles, 2003; Bacallado et al., 2013). In contrast, predictive characteri... | https://arxiv.org/abs/2502.10257v2 |
mixed binomial point processes. See Section 2 for the definition of these processes. We specialize the general Bayesian analysis for some notable classes of priors, including Poisson, mixed Poisson, and mixed binomial processes. Our treatment and proofs heavily rely on point process theory and Palm calculus (pioneered ... | https://arxiv.org/abs/2502.10257v2 |
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